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AIP conference proceedings

Generalized spin squeezing criteria: Entanglement detection with collective measurements

Géza Tóth, Christian Knapp, Otfried Gühne, Hans J. Briegel, Alexander I. Lvovsky

Published Jan 1, 2009• 0 citations
Publisher

Abstract

What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized spin squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular momentum components. They can be used for the experimental detection of entanglement in a system of spin‐12 particles in which the spins cannot be individually addressed. We also show how to obtain the complete set of entanglement conditions if the mean spin and the spin correlation matrix are known. Finally, we apply our criteria for entanglement detection in spin models, showing that they can be used to detect bound entanglement in the thermal state of these systems.

DOI
10.1063/1.3131365

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What is the relation between spin squeezing and entanglement? To clarify this, we derive the full set of generalized-spin-squeezing inequalities for the detection of entanglement. These are inequalities for the mean values and variances of the collective angular-momentum components Jk{J}_{k}Jk​. They can be used for the experimental detection of entanglement in a system of spin-12\frac{1}{2}21​ particles in which the spins cannot be individually addressed. We present various sets of inequalities that can detect all entangled states that can be detected based on the knowledge of (i) the mean values and variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (ii) the variances of Jk{J}_{k}Jk​ in three orthogonal directions, or (iii) the mean values of Jk2{J}_{k}^{2}Jk2​ in three orthogonal directions, or (iv) the mean values and variances of Jk{J}_{k}Jk​ in arbitrary directions. We compare our inequalities to known spin-squeezing entanglement criteria and discuss to which extent spin squeezing is related to entanglement in the reduced two-qubit states. Finally, we apply our criteria for the detection of entanglement in spin models, showing that they can be used to detect bound entanglement in these systems.
We determine the complete set of generalized spin squeezing inequalities, given in terms of the collective angular momentum components, for particles with an arbitrary spin. They can be used for the experimental detection of entanglement in an ensemble in which the particles cannot be individually addressed. We also present a large set of criteria involving collective observables different from the angular momentum coordinates. We show that some of the inequalities can be used to detect k-particle entanglement and bound entanglement.
A complete set of generalized spin-squeezing inequalities is derived for an ensemble of particles with an arbitrary spin. Our conditions are formulated with the first and second moments of the collective angular momentum coordinates. A method for mapping the spin-squeezing inequalities for spin-12\frac{1}{2}21​ particles to entanglement conditions for spin-jjj particles is also presented. We apply our mapping to obtain a generalization of the original spin-squeezing inequality to higher spins. We show that, for large particle numbers, a spin-squeezing parameter for entanglement detection based on one of our inequalities is strictly stronger than the original spin-squeezing parameter defined in S\o{}rensen et al. [Nature (London) 409, 63 (2001)]. We present a coordinate system independent form of our inequalities that contains, besides the correlation and covariance tensors of the collective angular momentum operators, the nematic tensor appearing in the theory of spin nematics. Finally, we discuss how to measure the quantities appearing in our inequalities in experiments.
We derive conditions in the form of inequalities to detect the genuine NNN-partite entanglement of NNN systems. The inequalities are expressed in terms of variances of spin operators and can be tested by local spin measurements performed on the individual systems. Violation of the inequalities is sufficient (but not necessary) to certify the multipartite entanglement and occurs when a type of spin squeezing is created. The inequalities are similar to those derived for continuous-variable systems, but instead are based on the Heisenberg spin-uncertainty relation $\mathrm{\ensuremath{\Delta}}{J}_{x}\mathrm{\ensuremath{\Delta}}{J}_{y}\ensuremath{\ge}|\ensuremath{\langle}{J}_{z}\ensuremath{\rangle}|/2$. We also extend previous work to derive spin-variance inequalities that certify the full tripartite inseparability or genuine multipartite entanglement among systems with fixed spin JJJ, as in Greenberger--Horne--Zeilinger (GHZ) states and W states where J=1/2J=1/2J=1/2. These inequalities are derived from the planar spin-uncertainty relation ${(\mathrm{\ensuremath{\Delta}}{J}_{x})}^{2}+{(\mathrm{\ensuremath{\Delta}}{J}_{y})}^{2}\ensuremath{\ge}{C}_{J}$ where CJ{C}_{J}CJ​ is a constant for each JJJ. Finally, it is shown how the inequalities detect multipartite entanglement based on Stokes operators. We illustrate with experiments that create entanglement shared among separated atomic ensembles, polarization-entangled optical modes, and the clouds of atoms of an expanding spin-squeezed Bose-Einstein condensate. For each example, we give a criterion to certify the mutual entanglement.
We present criteria to detect the depth of entanglement in macroscopic ensembles of spin- j particles using the variance and second moments of the collective spin components. The class of states detected goes beyond traditional spin-squeezed states by including Dicke states and other unpolarized states. The criteria derived are easy to evaluate numerically even for systems of very many particles and outperform past approaches, especially in practical situations where noise is present. We also derive analytic lower bounds based on the linearization of our criteria, which make it possible to define spin-squeezing parameters for Dicke states. In addition, we obtain spin squeezing parameters also from the condition derived in (Sørensen and Mølmer 2001 Phys. Rev. Lett. 86 4431 ). We also extend our results to systems with fluctuating number of particles.